area ← dim ##.ksphere radius                ⍝ Hypersphere surface area.

Inspired by Eugene McDonnell's paper [1],  [ksphere] returns the surface area of
an ⍺-sphere of radius ⍵.

NB:  Wolfram [4] points out: "Unfortunately, geometers and topologists adopt in-
compatible  conventions  for the meaning of "n-sphere," with geometers referring
to  the  number  of coordinates in the underlying space ("thus a two-dimensional
sphere  is  a  circle,"  Coxeter 1973, p. 125)  and topologists referring to the
dimension of the surface itself". [ksphere] sides with the topologists so, here,
we call a circle a 1-sphere or S1.

Here are the surface areas  of some unit hyperspheres.  In particular, the "sur-
face area" of a 1-sphere (circle) is its circumference.

    (1 to 10) ksphere 1             ⍝ surface of unit ⍺-spheres.
6.2832 12.566 19.739 26.319 31.006 33.073 32.47 29.687 25.502 20.725

Notice that the area of a unit k-sphere achieves a maximum at  around  6 dimens-
ions. In his paper, Eugene poses the question: What dimension of space gives the
maximum hypervolume to the unit radius hypersphere?

As coded,  [ksphere] is a continuous function over real ⍺, so we can investigate
non-integral dimensions.  Here is a crude Newton-Raphson technique for finding a
local maximum.

    max←{⍺←⎕CT*÷2               ⍝ Local maximum using Newton-Raphson.
        ∆x←1+¯1 0 1×⍺           ⍝ x deltas.
        ⍺⍺{                     ⍝
            ∆∆x←⍵×∆x            ⍝ x-∊ x x+∊
            ∆∆y←⍺⍺ ∆∆x          ⍝ f(x-∊) f(x) f(x+∊)
            d1←÷⌿¯2-/↑∆∆y ∆∆x   ⍝ first difference.
            d2←÷/-/↑d1(2↑∆∆x)   ⍝ second difference.
            ∆←⌈/d1÷d2           ⍝ increment f'(⍵)÷f"(⍵)
            ∆=0:⍵               ⍝ approx convergence: done.
            ∇ ⍵-∆               ⍝ ⍵ → ⍵ - f'(⍵)÷f"(⍵)

        ksphere∘1 max 6         ⍝ approx maximum surface area of unit k-sphere.

Seaching the Internet shows the maximum to be at dimension 6.2569464048605768 to
17 sig figs. Wolfram Alpha [5] is particularly good for this sort of exercise.

Notice that the volume of the (k+1)-ball inside the k-sphere is  easily  derived
from its surface area:

    kvol ← {⍵×((⍺-1) ksphere ⍵)÷⍺}        ⍝ volume of ⍺-ball of radius ⍵.

So to answer Eugene's question:

    kvol∘1 max 5                ⍝ approx maximum volume of unit k-sphere.


    Visualising Hyperspheres
    In general,  we can construct n-sphere Sn by  gluing  together  the  surface
    (n-1)-spheres of two n-balls. Let's start with a familiar 2-sphere S2:

    Take a pair of 2-balls (flat circular discs) of thin rubber sheet  and care-
    fully glue their outside (S1) edges  together.  Then  inflate  the  enclosed
    space to produce a regular 2-sphere. A Flatlander [7], living on the surface
    of S2, would  perceive  it  as  an  unbounded,  though  finite,  2-universe.
    Ferdinand Magellan was in a similar situation as he explored  his finite but
    unbounded 2-sphere.

    Similarly,  S1  may  be  constructed by gluing the endpoints of a pair of 1-
    balls  (line segments)  together  and  bowing  the  lines outwards to form a
    circle.  A "Linelander",  travelling around the circle, would not notice the
    two joining S0 points.

    We "3-landers" could build a 3-sphere by gluing the 2-sphere boundaries of a
    pair of "adjacent"  3-balls  (though we'd need to borrow a little 4-space in
    which to do the job).  To  make it easier, let's don scuba gear and swim in-
    side one of a pair of massive 3-balls (regular 3D balls) of sea-water, which
    are suspended close to each other in 4-space.  After our 4-lander friend has
    glued  together the outer 2-sphere-boundaries of our 3-balls  (using special
    transparent sea-water adhesive) we can swim from our home hemiball, straight
    ahead  in  any  direction,  into  the  other hemiball. If the sea-water glue
    really is transparent,  we should not notice as we swim through the 2-sphere
    join.  Again,  we're  in an unbounded manifold:  even though we are within a
    finite volume of water, no matter how far we swim in any direction,  we will
    never encounter a boundary.

    NB: Before  using  a harpoon gun, please heed the warning, at the end of the
    notes on →life←, about using artillery in a finite manifold.

    The Poincaré Conjecture
    Imagine a 2-dimensional (Flatland) spider wandering around in the surface of
    a large soap bubble.  To  amuse  herself in this bleak landscape she plays a
    little game:  as she moves, she extrudes a single filament of web, which is,
    of course, also embedded in the surface of  the  S2  bubble.  Her game is to
    roam around her 2-sphere looking for the starting end of the  filament  and,
    when she finds it, to reel it in.  As  she  is holding both ends of the fil-
    ament, topologically speaking, it forms a circle S1.  Our  spider finds that
    she can _always_ reel in her web-loop  and so she convinces herself that she
    must be in the surface of a sphere, rather than, for example, in the surface
    of a more exotic 2-manifold, such as a torus (S1×S1).

    We can play the same game in our sea-water 3-sphere:  we swim forwards while
    uncoiling our wreck-diving rope,  leaving one end at a fixed position in the
    3-water.  We find that, after swimming straight ahead in any direction for a
    distance of 2×D, where D is the diameter of our 3-spheres, the  end  of  the
    rope hoves into view again.  Now, if we grab both ends of the rope and start
    to pull, we should be able to reel in the whole of the loop without its  be-
    coming tight.

    By analogy with the flat-spider in the surface of torus S1×S1, we might  not
    always be able to reel in our rope had we found ourselves swimming,  for ex-
    ample inside S1×S1×S1,  which is made by gluing together opposite faces of a
    _cube_ of sea-water.

    It has long been assumed that a 3-sphere is the only 3-manifold in which one
    can always reel in the rope-loop in this way.  This  assumption  is known as
    the "Poincaré Conjecture" and a proof of it,  which carried a million-dollar
    reward, eluded mathematicians for the whole of the  twentieth  century.  The
    conjecture was proved in 2002 by Grigori Perelman, building on work by Rich-
    ard Hamilton. See ref[8] below.

    Explore 3-manifolds by downloading this magnificent hyper-flight-simulator:

    (muse: At a picnic, never let a topologist slice the loaf.)

Technical note:

As [ksphere] utilises only scalar pervasive functions,  it is itself scalar per-
vasive. This means that it may be applied directly between conformable arguments
of higher rank and depth.





⍝ 1-sphere is a circle:

    1 ksphere 10                    ⍝ circumference of circle, radius 10.

    2×○ 10                          ⍝ compare with: 2×Pi×r.

⍝ 2-sphere is a reqular sphere in 3-space:

    2 ksphere 10                    ⍝ surface area of sphere, radius 10.

    4×○ 10*2                        ⍝ compare with: 4×Pi×r-squared.

    (0 to 10) ksphere 1             ⍝ surface of unit ⍺-spheres.
2 6.2832 12.566 19.739 26.319 31.006 33.073 32.47 29.687 25.502 20.725

    kvol ← {⍵×((⍺-1) ksphere ⍵)÷⍺}  ⍝ volume of ⍺-ball of radius ⍵.

    (0 to 10) kvol 1                ⍝ volumes of unit ⍺-balls.
2 3.1416 4.1888 4.9348 5.2638 5.1677 4.7248 4.0587 3.2985 2.5502 1.8841

See also: kball to life

Back to: contents

Back to: Workspaces